Contents

Idea

Nonstandard analysis is a rich formalization of analysis that uses a certain explicit notions of infinitesimal objects. In fact, not only infinitesimal but also infinitely large can be accomodated (and must be). Moreover, not only the field of real numbers, but more general algebraic structures can be extended, essentially via a construction of ultraproducts; also general sets can be extended to contain nonstandard elements (see internal set).

Motivation

At its beginning, infinitesimal calculus was developed nonrigorously, though many interesting arguments and formal manipulations were found. Cauchy and Weierstraß introduced the ϵ\epsilon-δ\delta approach, which enabled modern rigorous analysis, but sometimes this method is cumbersome. For example, sometimes one needs to work with several infinitesimal levels or kinds of continuity in the same problem, and finding estimates may be very cumbersome. One would like to introduce infinitesimal quantities as additional elements of the sets of usual (‘standard’) quantities. Several related rigorous frameworks appeared under the name of nonstandard analysis, since the first such discovered by Abraham Robinson. Most often, approaches using ultrafilters, certain classes called internal sets and using topos theory enable the foundation of nonstandard analysis. Many properties and theorems from classical analysis imply new statements of nonstandard analysis; the mechanism is the so-called transfer principle, which can be required axiomatically without respect to a particular model of nonstandard analysis.

Models

An approach via ultrafilters

Assuming the axiom of choice (whose full strength is not necessary), there exists a free (= not containing finite subsets) ultrafilterFF on the set of natural numbersℕ={1,2,3,…}\mathbb{N}=\{1,2,3,\ldots\}, and such ultrafilters are in 11–11 correspondence with finitely additive measures on ℕ\mathbb{N} (using the algebra of all subsets) taking values in the two element set {0,1}\{0,1\}.

Fix a free ultrafilter FF on ℕ\mathbb{N}, and consider the set of all sequences of real numbers, ℝℕ\mathbb{R}^{\mathbb{N}}. So an element in here is a sequence

a=(a0,a1,a2,⋯)
a = (a_0, a_1, a_2, \cdots)

of real numbers.

We write f∼Fgf\sim_F g if the set {i∈ℕ|f(i)=g(i)}\{i\in\mathbb{N}|f(i)=g(i)\} belongs to FF; these are precisely the sequences which are equal almost everywhere with respect to the associated measure. The relation ∼F\sim_F is an equivalence relation and *ℝ:=ℝℕ/∼F{}^*\mathbb{R} := \mathbb{R}^{\mathbb{N}}/{\sim_F} is a nonstandard extension of ℝ\mathbb{R}, whose elements are sometimes called hyperreal numbers.

(This is a special case of the ultraproduct construction in model theory. In fact, we could have started with an ultrafilter on any set, not just ℕ\mathbb{N}. Such more general ultraproducts are necessary in order to obtain more refined models of nonstandard analysis satisfying stronger “saturation” principles.)

Given f∈ℝℕf\in \mathbb{R}^{\mathbb{N}}, we write fFf_F for its equivalence class in *ℝ{}^*\mathbb{R}. In particular, given any real number r∈ℝr\in \mathbb{R} the image *r:=(i↦r)F^* r :=(i\mapsto r)_F of the constant sequence

r=(r,r,r,⋯).
r = (r,r,r, \cdots )
\,.

is an element in *ℝ^*\mathbb{R} and this gives an injection*:ℝ↪*ℝ*:\mathbb{R}\hookrightarrow {}^*\mathbb{R}.

This is indeed an extension of uu in the sense that *u(*r1,…,*rn)=*r^* u({}^*r_1,\ldots,{}^*r_n)={}^* r iff u(r1,…,rn)=ru(r_1,\ldots,r_n)=r. This way, the usual operations +,⋅+,\cdot and the absolute value |⋅||\cdot| extend to *ℝ^*\mathbb{R}; usually one denotes these and other standard operations on *ℝ{}*\mathbb{R} without putting *^* in front, writing simply e.g. fF+gFf_F+g_F.

To extend division appropriately, we need a little bit more care as it is originally just partially defined, so we need an extension of the formalism to subsets of the real line. In particular there is a following definition of an extension *E⊂*ℝ^* E\subset{}^*\mathbb{R} of a subset E⊂ℝE\subset\mathbb{R}:

(For example, a number is positive, as defined earlier, if an only if it belongs to *{r|r>0}^*\{r|r \gt 0\}.) Then division is extended to a function *ℝ×*{r|r≠0}→*ℝ^*\mathbb{R} \times ^*\{r|r \ne 0\} \to ^*\mathbb{R}. If 1/x1/x is infinitesimal, then xx itself is infinite.

Conversely, an element x∈*ℝx\in{}^*\mathbb{R} is finite if |x|<*r|x|\lt {}^* r for some r∈ℝr\in\mathbb{R}. Every finite element x∈*ℝx\in{}^*\mathbb{R} is infinitely close to a unique real number q∈ℝq\in\mathbb{R} in the sense that x−*qx-{}^*q is infinitesimal. We say that qq is the standard part of xx and is denoted by q=st(x)q= st(x). Given a real number r∈ℝr\in\mathbb{R}, the subset μ(r)\mu(r) of all elements x∈*ℝx\in{}^*\mathbb{R} such that st(x)=rst(x)=r is said to be the monad of the real number r∈ℝr\in\mathbb{R}. Monads should be thaught of as infinitesimal neighborhoods. An elementary fact: a subset E⊂ℝE\subset\mathbb{R} is open iff μ(r)⊂*E\mu(r)\subset{}^*E for all r∈Er\in E; EE is closed iff st(x)∈Est(x)\in E for all finite x∈*Ex\in{}^* E; and EE is compact iff, for all x∈*Ex\in{}^* E, xx is finite and st(x)∈Est(x)\in E.

In this model of nonstandard analysis, the transfer principle? is a corollary of a general theorem on ultraproducts due Jerzy Łoś. It can be stated in terms of a certain formal language L(ℝ)L(\mathbb{R}) of the real numbers. We can also extend this model to ultrapowers of larger sets, not just ℝ\mathbb{R} itself, with a corresponding extension of the language. In the limit where we reach an entire “universe” of mathematics, this leads to the topos-theoretic filterquotient and sheaf models below.

Filterquotients of topoi

The ultrapower construction above can be performed in the general context of topos theory. From any topos ℰ\mathcal{E} and any filter Φ\Phi of subterminal objects in ℰ\mathcal{E}, one can construct a topos ℰ/Φ\mathcal{E}/\Phi, the filterquotient? of ℰ\mathcal{E} by Φ\Phi. There is a logical functorℰ→ℰ/Φ\mathcal{E} \to \mathcal{E}/\Phi.

If ℰ=Set/ℕ\mathcal{E} = Set / \mathbb{N}, then any filter on ℕ\mathbb{N} gives a filter of subterminals in ℰ\mathcal{E}, whose corresponding filterquotient corresponds to the above construction. The composite functor

Set→Set/ℕ→(Set/ℕ)/Φ Set \to Set/\mathbb{N} \to (Set/\mathbb{N})/\Phi

might be written *(−)^*(-). If Φ\Phi is an ultrafilter, then (Set/ℕ)/Φ(Set/\mathbb{N})/\Phi is a two-valued topos, whose internal logic is essentially that of the ultrafilter model described above. In particular, the global elements of *ℝ^*\mathbb{R}, as an object of this topos, are precisely the “hyperreal numbers” described above.

In this context, the transfer principle? is the fact that the functor *(−)^*(-) is both logical and conservative, and hence it both preserves and reflects the truth of formulas in the internal languages.

Sheaves on the topos of filters

A different topos-theoretic construction is to consider the topos of sheaves on a category of filters. This topos models the internal set theory of Nelson, a more axiomatic approach to nonstandard analysis. References:

Measure and generalized functions

The Lebesgue measure on Rn\mathbf{R}^n extends to Loeb measure on *Rn{}^\ast\mathbf{R}^n. This may be used for probability theory and also for generalized functions.

The theory of generalized functions of Schwarz can be reproduced by nonstandard analysis:

Theorem. (Abraham Robinson) Every generalized function f:R→Rf:\mathbf{R}\to\mathbf{R} can be represented as the integration of the product of test function with a nonstandard smooth function f˜:*R→*R\tilde{f}: {}^\ast\mathbf{R}\to{}^\ast \mathbf{R}

(f,ϕ)=∫f˜ϕdx
(f,\phi) = \int \tilde{f} \phi d x

There is also inutionistic version of nonstandard analysis approach to generalized functions as well as nonstandard approaches to Sato hyperfunctions (Sousa pinto), to Coulombeu distributions etc.

While the most common infinitesimals appearing in SDG are nilpotent, in contrast to those of NSA which are invertible, some models of SDG do contain invertible infinitesimals; see here.

Since the surreal numbers are the universally embedding ordered field, any field of hyperreals can be embedded in the surreals. However, such embeddings don’t seem very useful, since they don’t preserve any of the important structure of the hyperreals (such as the transfer principle).